Well, the first question is why do you NEED to use the ESC as part of the binding equation? Unless I'm missing something, it should not matter, you're not binding the RX and ESC together but rather the RX and radio. You can simply plug a battery into the RX for binding purposes. You only need the ESC plugged into the RX to perform the throttle calibration and actual useage.

You don't mention what new ESC you're using so this begs the question: What is the output BEC voltage of the new ESC? Is it possible you fried the RX? This is clearly has nothing to do with the radio if the RX won't even blink with the bind plug plugged in to indicate it's in bind mode. Unless the bind plug pins are in the wrong location (should be on the outside ends), this one is a doozie.